In a medium bowl, whisk together the all-purpose flour, whole wheat flour, baking powder, baking soda, and salt, then set aside.
Using a hand mixer or a stand mixer fitted with the paddle attachment, whip together the softened butter, sugar, and light brown sugar in a large bowl until light and fluffy, around 3 minutes.
Mix in the peanut butter, followed by the egg white and vanilla extract. Once evenly combined, stir in the dry ingredients until a dough forms.
Wrap the dough in a square of plastic wrap and flatten to an approximate ¾'' thickness, then chill for at least 1 hour.
After an hour, unwrap the cookie dough and allow it to sit at room temperature for 5 minutes. In the meantime, preheat the oven to 350°F.
Place the dough onto a lightly floured work surface, then roll to ¼ to ⅛'' thickness. If desired, use a cookie stamp to emboss the surface of the dough with a Nutter Butter design, then use a cookie cutter to stamp out individual cookies.
Arrange the stamped cookies on a baking sheet lined with parchment paper, leaving about ¼ inch gap between each one. Bake for 12-15 minutes, until the surface of the wafers are firm, dry, and lightly golden around the edges.
Let the cookies cool to room temperature before assembling. In the meantime, prepare the frosting.
To make the peanut butter filling, beat the softened butter, peanut butter, and honey in a bowl until smooth. Add in the powdered sugar a little at a time, followed by the vanilla and salt.
Mix at a medium-high speed until the frosting is light and fluffy. If the buttercream is too thick, add milk one tablespoon at a time until the desired consistency is reached. Transfer the filling to a piping bag fitted with a nozzle attachment.
When ready to assemble, flip half of the cookies over. Pipe a layer of peanut frosting in the center of each wafer, then sandwich with the remaining cookies. Allow the filling to set completely before serving.
Leftover cookies can be stored in an airtight container at room temperature for up to 1 week or in the freezer for 3 months. Serve at room temperature.